Right at the west end of the sea wall is Dellanera RV Park. It is rather tight, but right on the beach. We have stayed here numerous times. You didn't mention what type of RV you are traveling in, but they also have pull thrus, but you should make a reservation.
A bit farther out is Galveston Island State Park. If you get there early on a weekday you might be able to get a front row site next to the beach. This is also a nice park. Covered shelters with picnic tables. If you get the front row spot and you are coordinated, you can put your chair on the table top of the picnic table and get a perfect view of the Gulf over the dunes! I spent quite some time enjoying this one summer! Just be careful getting up and down.
Since Landry's owns most of the restaurants in Galveston, you would be wise to buy some fresh seafood somewhere on the island and take it back to the RV and cook it there! Make sure and take mosquito repellant! Hope you have a good time, we love Galveston!